16
1 Introduction
1.4 Quantum Theory of Large Systems
1.4.1 Models of systems with infinite number of degrees of freedom enter to quantum theory when we want to describe either processes accompanied with changes
of numbers of particles (resp. quasiparticles) present in the physical system (what
also occurs each time if we try to describe quantal analogues of classical continuous
media, resp. fields), or systems with actual infinity of particles (the ‘thermodynamic
limit’ necessary e.g. for clear conceptual description and abstract investigation of
phase transitions). In standard models of infinite systems in quantum theory the
algebras of bounded observables (e.g, CCR or CAR algebras for infinite number of
degrees of freedom or algebras of spin systems on infinite lattices) have many mutually unitarily inequivalent physically relevant representations as algebras of bounded
operators in some Hilbert spaces. These inequivalent representations might correspond e.g. to various states on the algebra of observables representing situations
with various values of some macroscopic–global parameters of the large syatem.
It often happens, moreover, that for description of some processes (time evolution,
symmetry transformations), we are not able to work in the framework of only one
(even faithful) representation. It is, consequently, useful to formulate theoretical
scheme for the quantum theory of large systems (QTLS) in a representation independent, algebraic language. As basic sources of most of the here necessary mathematics and its application to description of large quantal systems could be taken, e.g.
[53, 54, 84, 223, 235, 274, 286, 289]; a very brief summary can be found also in
[37, Sect. 3.4].
1.4.2 A C
∗ -algebra A [37, B.2] (details on C
∗ -algebras can be found in [53, 54, 90,
91, 106, 235, 274, 275, 305, 306]) corresponds to any physical system in QTLS.
A is a Banach algebra over complex numbers with involution x → x
∗
, x ∈ A, and
with special (C
∗ ) property. This means that it is a norm-closed linear space endowed
with associative and distributive multiplication, and for any x, y ∈ A, λ ∈ C, and
with x ≥ 0—the norm of x ∈ A, it is: x y ≤ ≤x · ·y, the involution x → x
∗
is antilinear: (x + λy)
∗
= x
∗
+ λy
∗ , where λ is the complex conjugate of λ, with
(x y)
∗
= y
∗ x
∗ , x
∗
= =x (= 0 iff x = 0), and the C
∗ -property means: x
∗ x =
x
2
, ∀x ∈ A. A is called unital C
∗ -algebra if it contains unit element e ∈ A : ex =
xe = x, ∀x ∈ A. Selfadjoint elements x = x
∗
∈ A represent bounded observables
of the system. The algebra A is the algebra of observables of the system. For many
interesting systems, A is constructed as a C
∗ -inductive limit of a net of local algebras
of finite (sub)systems (see [53, 106, 274], and specifically [274, 1.23]); in this case,
these finite systems are interpreted e.g. as systems located in bounded space (-time)
regions. Quasilocal algebras used in QTLS have such a structure (see [53, Definition
2.6.3]). It will be useful in our considerations to connect the quasilocal structure of
A with an action of a (usually abelian) group ( is an infinite set—it might
be a locally compact noncompact group) on A: For any p ∈ let π( p) ∈
∗ - Aut
A, π( p 1 p 2 ) = π( p 1 )π( p 2 ) (p 1 , p 2 ∈ ). Let act transitively on a noncompact
locally compact space V and let to any bounded open subset v ⊂ V (denote the
set of all such subsets by B(V )) corresponds a C
∗ -subalgebra A v of A, the local
1 Introduction
1.4 Quantum Theory of Large Systems
1.4.1 Models of systems with infinite number of degrees of freedom enter to quantum theory when we want to describe either processes accompanied with changes
of numbers of particles (resp. quasiparticles) present in the physical system (what
also occurs each time if we try to describe quantal analogues of classical continuous
media, resp. fields), or systems with actual infinity of particles (the ‘thermodynamic
limit’ necessary e.g. for clear conceptual description and abstract investigation of
phase transitions). In standard models of infinite systems in quantum theory the
algebras of bounded observables (e.g, CCR or CAR algebras for infinite number of
degrees of freedom or algebras of spin systems on infinite lattices) have many mutually unitarily inequivalent physically relevant representations as algebras of bounded
operators in some Hilbert spaces. These inequivalent representations might correspond e.g. to various states on the algebra of observables representing situations
with various values of some macroscopic–global parameters of the large syatem.
It often happens, moreover, that for description of some processes (time evolution,
symmetry transformations), we are not able to work in the framework of only one
(even faithful) representation. It is, consequently, useful to formulate theoretical
scheme for the quantum theory of large systems (QTLS) in a representation independent, algebraic language. As basic sources of most of the here necessary mathematics and its application to description of large quantal systems could be taken, e.g.
[53, 54, 84, 223, 235, 274, 286, 289]; a very brief summary can be found also in
[37, Sect. 3.4].
1.4.2 A C
∗ -algebra A [37, B.2] (details on C
∗ -algebras can be found in [53, 54, 90,
91, 106, 235, 274, 275, 305, 306]) corresponds to any physical system in QTLS.
A is a Banach algebra over complex numbers with involution x → x
∗
, x ∈ A, and
with special (C
∗ ) property. This means that it is a norm-closed linear space endowed
with associative and distributive multiplication, and for any x, y ∈ A, λ ∈ C, and
with x ≥ 0—the norm of x ∈ A, it is: x y ≤ ≤x · ·y, the involution x → x
∗
is antilinear: (x + λy)
∗
= x
∗
+ λy
∗ , where λ is the complex conjugate of λ, with
(x y)
∗
= y
∗ x
∗ , x
∗
= =x (= 0 iff x = 0), and the C
∗ -property means: x
∗ x =
x
2
, ∀x ∈ A. A is called unital C
∗ -algebra if it contains unit element e ∈ A : ex =
xe = x, ∀x ∈ A. Selfadjoint elements x = x
∗
∈ A represent bounded observables
of the system. The algebra A is the algebra of observables of the system. For many
interesting systems, A is constructed as a C
∗ -inductive limit of a net of local algebras
of finite (sub)systems (see [53, 106, 274], and specifically [274, 1.23]); in this case,
these finite systems are interpreted e.g. as systems located in bounded space (-time)
regions. Quasilocal algebras used in QTLS have such a structure (see [53, Definition
2.6.3]). It will be useful in our considerations to connect the quasilocal structure of
A with an action of a (usually abelian) group ( is an infinite set—it might
be a locally compact noncompact group) on A: For any p ∈ let π( p) ∈
∗ - Aut
A, π( p 1 p 2 ) = π( p 1 )π( p 2 ) (p 1 , p 2 ∈ ). Let act transitively on a noncompact
locally compact space V and let to any bounded open subset v ⊂ V (denote the
set of all such subsets by B(V )) corresponds a C
∗ -subalgebra A v of A, the local